On De Giorgi's conjecture in dimension N>9

A celebrated conjecture due to De Giorgi states that any bounded solution of the equation u + (1 u 2 )u = 0 in R N with @yNu > 0 must be such that its level setsfu = g are all hyperplanes, at least for dimension N 8. A counterexample for N 9 has long been believed to exist. Starting from a minimal graph which is not a hyperplane, found by Bombieri, De Giorgi and Giusti in R N , N 9, we prove that for any small > 0 there is a bounded solution u (y) with @yN u > 0, which resembles tanh A t p 2 a , where t = t(y) denotes a choice of signed distance to the blown-up minimal graph := 1 . This solution is a counterexample to De Giorgi’s conjecture for N 9.

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